On quasi one or two dimensional superconductors

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On quasi one or two dimensional superconductors

J. Friedel

To cite this version:

J. Friedel. On quasi one or two dimensional superconductors. Journal de Physique, 1987, 48 (10),

pp.1787-1797. �10.1051/jphys:0198700480100178700�. �jpa-00210620�

On quasi ^{one or two} dimensional superconductors

J. Friedel

Laboratoire de Physique ^des Solides, bâtiment 510, Université Paris Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Requ ^{le 24} juin 1987, accepté ^{le 27} juillet 1987)

Résumé.

On souligne qu’un optimum doit être recherché entre les forces de couplage ^{inter et} intrablocs. Cet

optimum peut ^{être réalisé en} géométrie quasibidimensionnelle. ^Les oxydes supraconducteurs ^{à hautes} Tc ^sont discutés de ce point ^{de vue.}

Abstract.

One stresses that, ^at least in the weak coupling limit, ^an optimum ^must be struck between inter and intrablock coupling strengths. ^This optimum ^can be realised in quasibidimensional geometry. ^The supraconductive oxides with high Tc’s ^are discussed from this point ^{of view.}

Classification

Physics ^Abstracts

74.20

Introduction.

It is well known that the proper three dimensional

superconductivity vanishes in samples ^{of one or two} dimensions, owing ^to thermal fluctuation of the

superconductive ^order parameter [1, 2]. ^{A mean}

field critical temperature Ti (i ^{= 1,} 2) ^{can neverthe-}

less be computed, ^{which measures the local} strength

of superconductive couplings. ^{In the weak} coupling limit, ^and assuming ^an average effective coupling ^V

between electrons, ^this temperature is related to the

density ^{of states} n(E) by ^the implicit ^relation [3],

valid for infinite systems

where u ^{is the} Fermi level and wD a ^cut off pulsation (Debye frequency ^for phonon ^mediated supercon-

ductivity).

In one or two dimensions, n (E) ^has diverging

van Hove anomalies [4]. ^{When u is near to such an} anomaly, ^the explicit variation of n with E must be taken into account [5-7]. ^The integral (1) ^no longer depends ^{on its limits ±} hWD. ^{The «} classical » solu- tion of (1), valid in three dimensions

must be replaced by ^an expression ^{with a maximum}

value given approximately by

and

respectively, where til ^{is an} intrablock transfer inte-

gral, ^{related to} the intrablock band width (cf.

appendices A, B). ^It is then clear that, ^for given

values of the constants such that

the intrablock coupling strength, ^measured by kB Ti, ^{increases with} decreasing dimensionality.

If now one establishes a weak superconductive coupling between such blocks, ^one expects ^{a three} dimensional superconductivity ^{to be} produced ^below

a temperature Tc, ^{at most} equal ^to Ti. ^In fact, ^{it is}

well known that the entropy ^effects which alter

superconductivity ^above Tc ^{become more intense}

when the dimensionality of the blocks decreases [1, 2].

The condition of maximum Tc ^{as a function of the} blocks’dimensionality ^is therefore obtained by ^a

balance between antagonistic tendencies. _{This ques-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100178700

tion is studied here in a very qualitative way, and the

new oxides with hight Tc’s briefly ^{discussed in this} light. Equivalent problems ^of optimisation ^{would be}

met for other types of intrablocks superconductive couplings. ^The special properties ^of Tc ^and Ti ⁱⁿ quasi ^two dimensions is finally ^commented upon.

1. Optimal dimensionality in the limit of weak interb- locks couplings.

We consider here the simplest ^{cases of} parallel

chains or parallel planes ^{of atoms} within which the conductive electrons are strongly delocalized owing

to a transfer integral ti , ^{and with} relatively ^weak

effective interactions V. We also consider the limit where the electrons can jump between such neigh- bouring ^{« blocks », but with an} interblock transfer

integral t 1. ^{weak with} respect ^{to the other} par- ameters. Thus

For weak enough t .1’s, one can assume that their existence does not alter much the electronic structure of each block, ^and especially their intrablock super- conductive coupling energy 2 kB Ti. ^{One can also}

assume that the main superconductive coupling

between blocks is due to the exchange ^of Cooper pairs ; ^the interblock superconductive coupling ^ener-

gy is then given very roughly by [8]

where tl ^{is the} probability ^of exchange ^{of a} cooper

pair, ^{and 2} kB Ti ^its coupling energy within ^a block.

The obvious condition for this regime ^is

And expressions (3) ^and (4) ^for Ti ^{show this to be}

realized if indeed t-L ^{is small} enough compared ^{to V}

and tll , ^{as stated in} (6).

Classical relations give ^{then the} three dimensional critical superconductive temperature Tc.

In a quasi ^one dimensional situation, ^{one has} [1]

In a quasi ^two dimensional situation, ^{a similar} argument [2] ^{leads to}

The maximum values (3) ^and (4) ^then give

Comparing (2), (11) ^and (12) ^{in the limit} (6), ^the optimum T, ^{will be met for} quasi bidimensional

compounds, ^if (6) ^is satisfied. Furthermore, ^as

shown in appendices ^{A and} B, the actual value of

Ti, ^{thus of} Tc, will be less sensitive to deviations of the position of the Fermi level u ^{from its} optimum position ^{on a van Hove} anomaly ^{in two dimensions}

than in one dimension. Thus quasi bidimensional

compounds ^{can have} large Tc ^{over a broader} range of compositions ^than quasi ^one dimensional ones.

2. More detailed discussion on interblock couplings.

We wish first to elaborate on condition (8), ^then point ^{out some} specific effects in two dimensions, expected for very weak t 1- .

For a perfect crystal ^at 0 K, t, introduces a

modulation of the Fermi surfaces of amplitude

t-L along the directions of transfer [9]. ^{For chains} along ^{0 z, the} initially planar ^Fermi surfaces normal to 0 z are then slightly warped, ^{as shown in} figure ^{la. For} planes ^{normal to 0 z, the} initially cylindrical Fermi surface parallel ^{to 0 z is} slightly

modulated as shown in figure ^lb.

Fig. 1.

Warping of the Fermi surface due to t 1- : a) ^atomic chains ; b) ^atomic planes.

In the more frequent ^{case of a} crystal ^at high enough temperature ^{or with} enough ^static imperfec- tions, ^the frequency of relaxation of the electrons

along ^{the blocks can lead to a «} Dingle temperature ^»

hw R comparable ^{to or} larger ^than tl. ^{The interblock} electronic jumps then become diffusive [10] ^and

broaden the Fermi energy by ^{an amount of order} hWR -- t-L (Fig. ^2a, b). ^{In both cases, the effect of}

such a warping ^or broadening ⁱⁿ computing Ti ^is negligible ^as long ^as

a condition which generalises ^condition (8).

A second remark which needs be made for the

following discussion concerns the exact nature of the

fluctuation regime ^above T,, ^for quasi ^{two dimension-}

Fig. ^2.

Broadening of the Fermi surface due to a

Dingle temperature larger than tl : a) ^atomic chains ; b) ^atomic planes.

al compounds. It is indeed well known that, ^between Tc ^{and a critical} temperature Tc’ ^{near to} T2, ^the

thermal fluctuations introduce some compression ^or

dilatation of the superconductive ^order parameter

along ^the planes, ^without breaking ^{the order within}

the planes : ^{vortex lines are thus} introduced between the superconductive blocks. It is only ^{above the}

Kosterlitz and Thouless critical temperature Tc ⁼ T2

that the superconductive order breaks down, ^within

each plane, by the introduction of vortex lines

piercing ^the planes [11]. Thus between T, ^and Tc’ ^~ T2, ^one expects large superconductive ^fluc- tuations, which should give ^an extra current strictly parallel ^{to the} planes [12] ; ^if Tc ^is small, ^{this current}

should be strongly ^non linear and disappear ^{above a}

small critical current where vortex lines can be

multiplied ^between planes. Indeed in this case

Tc’ ^{is the} temperature ^of phase transition and

Tc only ^{a crossover} temperature.

3. The La2CU04 family.

The compounds ^of type La2Cu04 and substituted

(Sr, ^{Ba for} La) ^{have a structure of} parallel ^conductive CU02 planes, where electrons can jump ^{from one}

d(x2 - y2) orbital of a Cu atom to that of a neighbour through ^{an effective transfer} integral via the p orbitals of intermediary ^{0 atoms} [13]. In the limit where the 0 p orbitals ^are normally occupied ^and

this transfer is treated as a perturbation, ^{the band}

structure of independent ^electrons reads, ^{in a} tight binding approximation [14]

where Ed is the energy of the d (x2 - y2) copper

orbital, ^{a and b the distances to} its copper neighbours

in the CU02 plane, ^and

where _{’T a} is the transfer integral between the d and p orbitals and _so the energy difference them (Fig. 3).

In La2Cu04, the band is just ^half filled ; Cu, ^{Sr or Ba} doping ^increases the number of d holes in the band.

0 vacancies introduce supplementary ^electrons

which can tend to shift the Fermi level the other way. However when created in the CU02 planes, they ^{also scatter} very strongly the conductive elec- trons and can lead to localization effects. Other copper d electron occupy d ^states (xy, yz, ^zx,

322 - rfi ^{which are more} stable either because they

do not point ^towards neighbouring oxygens ^or because the CuO bond lengths ^{normal to the} CU02 planes ^are strongly elongated ^with respect ^to those in the planes.

Fig. ^3.

Structure of the CU02 planes (points, Cu ; circles, oxygens).

In the tetragonal phase (a ⁼ b ), ^the density ^of

states n(E) ^{has a} logarithmic ^{van Hove} anomaly,

which is split ^{into two in the} orthorhombic phase (a =F b ) (Figs. ^{4 and} 5). ^For purely stoechiometric

La2Cu04, ^this splitting necessarily lowers the elec- tronic energy, ^{and the amount of} splitting 4 ^is

obtained by balancing the energy gained by ^this

Fig. ^4.

^{Band structure in the} tetragonal phase ^of La2Cu04 : a) surfaces of constant energy ; b) density ^of

states.

Fig. ^5.

^{Band structure} in the orthorhombic phase ^of La2Cu04 : a) Surface of constant energy ; b) density ^of

states.

band Jahn Teller effect, ^{of order} A £2 In ê, by ^the

elastic reaction of the lattice, ^{of order} BE 2, ^if

E ⁼ A/8 ^{t. Hence at} equilibrium

Now taking

the energy splitting is of order

With [ 13] b a a -1°, ^{a qo a = 6, and [15] t == 0.8 eV,}

A should be of the order of 15 x 10- 2 eV.

For non stoichiometric compounds, ^{where the}

Fermi level deviates from the middle of the band,

one expects this Jahn Teller splitting ^{to decrease} progressively ^and disappear ^{when the} Fermi level falls somewhat (beyond) the energy band Ei ^to E2 ^{of width 2l} of maximum splitting [16]. ^{For a Fermi} level u ^below Ei the energy gained by ^{the Jahn}

Teller splitting ^{is of order} AE 2In 4t - ii ^{4t (cf.}

Appendix C). ^{This is} larger than the elastic energy

Be 2 if In 1 4 t 4 t ^{A B Comparing with the value}

e = 8 t ^{8t for undoped samples, as given by (16), one}

sees that the Jahn Teller splitting disappears ^if

From the density ^{of states} given ⁱⁿ appendix B, ^such

a shifting of the Fermi level should occur in

This is indeed the order of magnitude ^{of the} doping actually required ^to preserve the tetragonal phase ^at

all temperatures [16, 17] which is somewhat above

x = °2 y m 10 9E (Fig. 6).

Fig. 6. - Phase diagram ^for La1-xSrxCu04 - y: ^{0 or-}

thorhombic phase ; ^t tetragonal phase ; ^s superconductive phase.

Exact nesting conditions occur for undoped La2CU04, whether in the tetragonal ^{or in the or-}

thorhombic phase (Figs. 4, 5). ^The corresponding nesting ^{vectors are} Q ⁼ 7r 7r _{a b} ^{For the exactly}

stoichiometric compound, ^{one then} expects ^{a further} instability, ^{which can be a} priori ^{either a lattice or a} spin modulation, depending ^on the relative strengths

of the corresponding electron-phonon ^{and electron-}

electron couplings [6, 7, 13, 14, 16]. Whatever the exact nature of such a modulation, ^{it should} disap-

pear rapidly ^with doping, ^{because the} nesting ^condi-

tions are quickly becoming much less favourable when the Fermi energy deviates from the middle of the band. A recent computation [18] predicts ^{that a} doping x - 2 y of about 1 % is enough ^{to kill this}

modulation if it was made for the tetragonal phase.

Appendix ^D discusses the case of the orthorhombic

phase. Figure ^{6 then} gives ^the predicted phase diagram, ^{in fair} agreement ^with experiment. ^There

seems indeed to be a SDW phase ^{for x -} 2 y ¹ %, although ^its exact nature is not known [19, 20]. ^In particular, it can’t be a simple ^3d antiferromagnetic

arrangement ^{of vector} Q ^{in the} Cu02 planes, ^{as each}

Cu of one plane ^is equally coupled with 4 Cu of a

neighbouring plane.

It is on a phase diagram ^{such as} figure ^{6 that} superconductivity ^{must be} discussed. The ratio

t .L/tll is of order 10-1, ^as deduced from the ani- sotropy ^of resistivity ^{in such} compounds [21]

Equation (12) ^{then explains a maximum Tc == 40 K if}

y - ^0.8, which falls indeed just ^within the weak

2t ll

coupling ^limit. As, when the Fermi level u ^deviates

from the van Hove anomaly Ei by ^{more than the}

superconductive gap 5, T2, ^{and hence} Tc decrease in

the ratio ð / I J.L - Ell, ^{, one expects to observe a}

maximum of Tc in ^the orthorhombic phase, ^{when the}

Fermi level should be at the van Hove anomaly Ei, ^{and decrease} slowly ^on either side (Fig. 6). ^This

is indeed observed [16]. ^{The value}

of the maximum T, is indeed that computed ^for

u ⁼ El ^{with the same} parameters ^{as for}

equation (20).

Undoped compounds ^{can exhibit} superconductivi-

ty with sizeable critical temperatures [16, ^22, 23].

However these are either made from too high ^a proportion of copper (1 ^{to 4} %) which could act as

dopant [22], ^or they ^are very inhomogeneous ⁱⁿ

oxygen vacancies, ^{as shown} by their very weak Meissner effect and coexistence with the SDW phase [16, 23]. There does not seem therefore to be any clear cut objections ^{to a} phase diagram ^{such as} figure ⁶ except ^that large concentrations of 0 vacan-

cies seem to introduce complex localization effects

[16, ^{22, 23,} 24].

4. The YBa2CU307 family.

This case is slightly ^more complex. Tc ^{is known to}

decrease fast with 0 deficiency [25, 26] ; ^{and the}

exact distribution of the oxygens is still ^{a matter} of

some controversy. ^{From a} fuller discussion [27], ^one

can extract the following points.

In the structure as usually proposed [28-30] ^rows

of parallel CuO chains between planes ^of CU02 ^build fairly strongly coupled « sandwiches » which are

weakly coupled through planes of Y ions. A fairly

strong T2 ^{can be} assigned ^{to each sandwich. But the} expected ^weakness of t-L through ^the planes ^{of Y and}

the insensitivity ^of superconductivity ^to replacing ^Y by magnetic ^{rare earths} [31, 32] ^could possibly ^lead

to a situation where the real three dimensional critical temperature Tc is small and what one ob-

serves is due to 2d superconductive fluctuations [12],

up ^to T2. ^{In that case,} the weakness of the critical

current could be intrinsic in a single crystal ^{and due}

to the ease of creating ^{vortex lines} along ^{the Y} planes. ^It could however be counteracted by ^a

suitable texture either finely polycrystalline ^{or in}

thin epitaxial layers ^with CU02 planes ^{at an} angle ^to

the layer ^{surface, or} by ^numerous defects in the

stacking ^{of the Y} planes [33, 34].

In a different structure proposed recently [35],

YO layers join sandwiches of CUO/CU02/CUO layers. ^{In the} superconductive orthorhombic struc-

ture, ^{the 0} of the YO layers allow electrons to delocalise along (square) CU02 planes ^{normal to the}

Y and Ba layers, ^{with a} preference for the Cu atoms

(A) ^{near to the Y} planes, ^{which should be} nearly Cu+ + , while the Cu atoms (B) ^{between BaO} planes

should be nearly ^{Cu+ + + .} Coupling between such

CU02 planes ^{would be} through empty ^{CuO chains}

normal to them and situated between BaO planes.

The considerations of this paper would apply ^most directly ^{to this second} type ^of structure. Despite ^the potential modulation of A and B atoms in the

CU02 planes, the van Hove logarithmic ^anomalies

characteristic of 2d geometry ^are preserved ^and

indeed multiplied, ^so that it would be reasonable to assume such an anomaly ^{to be} present ^{near the} Fermi level. And the presence of the CuO chains would provide ^a simple ^and fairly ^effective coupling

between the CU02 planes.

While the exact structure for the stoichiometric

compound YBa2Cu307 ^{seems rather} complex ^and

has not been solved completely by ^{neutron diffrac-}

tion [36], preliminary ^{studies seem more in favour of}

the first model, ^as also the first measurements of the

anisotropy ^of Hc2 ^on single crystals [37]. ^Also possible ^{lattice or} spin modulations due to nesting

condition of the Fermi surface have not been

thoroughly studied in this case.

5. Conclusions.

The special advantages ^{of a} quasi ^2d geometry ^for weak coupling superconductivity ^{have been} stressed,

and La2-xSrxCU04-y ^{could be a good example of}

the compromise ^{that has to} be struck between the intra and interblock couplings. ^{At the} present time,

different schemes are possible ^for YBa2CU307 along

similar lines, ^{but the most} widely accepted ^one suggests that the weak critical currents could possibly

be an intrinsic property ^of large perfect single crystals.

The weak coupling limit has been chosen in this paper because of the fairly large conduction band widths that are both computed ^and observed in these

compounds [38, 39].

One could imagine ^{that a similar} study ^{could be}

made in a strong coupling limit, although ^{in that case}

the intrablock superconducting coupling ^{should be} expected ^to be less sensitive to dimensionality.

Even in the weak coupling ^{limit, several} points

would be worth looking ^{in more details :}

-

It is not clear up till ^now whether the supercon-

ducting coupling ^{V is} through electron-phonon [14]

or electron-electron [6] (e.g. electron-antifer-

romagnetic spin ^waves [18, 41]) ^as also the assump- tion of a constant value of V. The actual role of the

van Hove anomalies depends ^{on this} [14].

-

In either case, ^as superconductivity ^{seems in} competition ^{with a lattice or} spin modulation induced

by nesting, ^{one has to} study ^the possible coupling

terms between the two instabilities [12, 14], ^{as done}

e. g. in organic superconductors [41].

-

Details of the band ^structure and thus numeri- cal estimates depend ^{on the} approximations ^made,

e.g. that conduction electrons concentrate on the Cu 3d states and hopping through ^{0 is} only perturbative.

Changing ^these assumptions ^{would not} fundamental7

ly change ^{the existence and nature of the van Hove} anomalies, except for very special ^and unlikely ^cases

where van Hove anomalies could become stronger than usual for the corresponding dimensionality.

However such changes ^{could alter} significantly ^the

details of nesting conditions, thus the conditions for the lattice or spin instabilities discussed for

La2-xSrxCuO4-y’

-

It is clear that, ^besides playing ^{on the} filling ^of

the d band, the presence of 0 vacancies can lead to

strong localisation effects when they ^{occur in the} Cu02 planes. Furthermore these 0 vacancies are not

distributed uniformly ^{in the} grains ^{of a} sample prepared ^{in «} usual » ways and their concentrations and repartition ^are very sensitive to the details of treatments with temperature, temperature ^variations

and 0 pressure. There is obviously ^{here a dimension}

of the problem ^{which is} only ^now been attacked

seriously [16].

-

Finally, and this is at the moment the most crucial point, ^the strength of the interblock transfer

integral tl ^{is very} pooly known-indeed unknown in

YBa2CU307- Its effect on the interblock supercon..

ductive coupling ^{is also} expressed ^{in a} very rough

way. This is probably ^one of the facets of the

problem ^which requires ^most urgently ^{to be studied}

in depth.

It is also clear from this discussion that if a quasi

2d geometry ^has especial attractions for high Tc, ^{there are} many possible variations on this theme,

and therefore many conductive compounds ^{can be} potential candidates for still higher ^{values of} Tc.

The author acknowledges with thanks many early

discussions with J. Labbe, ^S. Barisic, ^{M. Rice and}

H. M. Schulz, ^{as well as more recent ones with}

P. Lederer.

Appendix A. Estimate of Tl ^{as a} function of the

position of the Fermi level with respect ^{to the ld van} Hove anomaly (CuO chains).

This problem has been solved exactly by ^{Labbe in}

another context [5]. ^We give ^{here a} simplified ^but approximate ^treatment.

For a chain of CuO, where each Cu has four

equidistant ⁰ neighbours, ^two along ^{the chain Ox}

and two in a direction y ^{normal to the chain, the}

d(x2 - y2) band of copper is given by

where the hopping through ^{the 0 atoms} is treated as a perturbation, and thus t is related the dp ^transfer integral and energy difference by equation (15). ^One

deduces for equation (1) ^a density ^{of states} per Cu atom and per spin

where Ei ⁼ Ed ^{+ 2 t is the} (lower) ^band edge ^{H the}

Heaviside function.

Equation (1) ^{then reads}

where

This equation ^can be solved approximately by replacing ^{tanh x} by x for x I ^{1 and} by ^{± 1 for}

x I > .

This gives

The corresponding variation of X J a with ul is

given ⁱⁿ figure (A.1) ^{for a «} UD- It shows a maxi-

mum value of X - 4.6 a - 1/2 _{for ul} slightly ^smaller

than - a. According ^to (A.3), ^it corresponds ^{to a}

maximum value T1 max of T1 ^of order given by equation (4). Figure A, shows that B/x ^{and thus} Til2 keep within 50 % of this maximum values for - 2 a u1 5 a , ^thus

Fig. ^A.1.

X(u1 ) for 1d chain.

Appendix ^B.

Estimate of T2 ^{as a} function of the

position of the Fermi level with respect ^{to a 2d van} Hove anomaly (CU02 planes) [7].

For the tetragonal phase, ^where tx ⁼ ty ^{= t, the} density ^{of states near the van Hove} anomaly ^where

E ⁼ Ed ^{+ 4 t} (Eq. (14) ^and Fig. 4) ^is given per Cu atom and per spin by

where

Writing

we obtain

In the orthorhombic phase, ^{each van Hove} singulari-

ty ^is given by ^{a similar} formula, ^{where the numerical}

factor is divided by two, ^as long ^{as the} splitting ^{A is}

small compared ^{with t.}

If we retain the dominating logarithmic ^{term in} (B.1), equation (1) ^{then reads, with the same nota-}

tions (A.4),

The same approximation about tanh x leads to

The maximum value of X is obtained for ul ⁼ 0 :

For a UD, the ^term in ln2 a dominates and leads

to equation (3) for the maximum value T2 ^{max of} T2.

For a u UD, ^{one can} write

A development of the first integral in the second

member near u ⁼ ul shows that the leading ^{term in} (B.5) and thus in X(ul) ^is

Thus, in this range

T2 ^{is thus reduced} to 1 ^of its value when 4

u 2013 El ^{~ 8} kB T2 ... - Comparison with appen- dix A shows that Ti decreases less rapidly ^{from its}

maximum value with a shift of the Fermi level from its optimum ^{value in two than in one dimension.}

Appendix ^C. Jahn Teller splitting ^{in the} tetragonal ^to

orthorhombic phase transition.

We compare the electronic energy per copper ^atom in the orthorhombic and tetragonal phases

with

With reduced variables

these expressions give for the orthorhombic phase

and similar expressions ^{for the} tetragonal phase.

For an exactly half-filled band (A = 4 t ), the energy change ^{reduces to}

where

When the Fermi level is well below the lower van Hove anomaly (u El ), ^a development ^of (C.1) ^and (C.2) gives

One deduces a shift in Fermi level given by

and an energy change

Comparison ^with (C.3) shows that In e I ^{is replaced} by ^an expression ^{where the} leading ^{term is} In 1- ¹¹ Mo! I ^{as long as} the Fermi level is not too far from the middle of the band.

In the superconductive phase, ^{these estimates}

should be somewhat altered by the presence of the

superconductive gap. Both the temperature ^{of the} orthorhombic-tetragonal phase transition and the amount of tetragonal distortion should be affected.

However these effects should be small, ^{as the} superconductive gap is small compared ^{with the}

internal energy involved in the phase change.

Appendix ^D. Stability ^{of a SDW in} La2 - xSr xCuO 4 - y.

The exact nature of a SDW in the La2Cu04 ^base compounds ^is unknown, ^{as the} dependence ^{of its}

detailed structure with doping. It is however reason-

able to assume that it involves a nesting ^vector Q = :t 7T , ^, :t 7Tb ^{in the CU02 planes.} The critical

a b

carrier concentration x - _{2 y} can then possibly ^be

deduced at T ⁼ 0 from the intraplane stability

criterion

-

where U is the electron-electron Coulomb interac- tion on a Cu site

the Fermi Dirac distribution function, ^{and the ener-} gies ^are counted from the middle of the d band.

From

it follows that

Hence (D.I) ^{reads, at 0 K}

where, ^{for the} orthorhombic phase where the SDW modulation seems to appear

If we introduce u ⁼ £k/4 t, (D.3) gives

A development ^{of the} integrand gives, ^if

u2 - U 1 > I /i I, ^and with u2 ^{= -} u1, the approximate

solution

to be compared ^with

for the tetragonal phase.

Condition (D.6) gives

and the corresponding ^{value of the} phase boundary

is

obtains, ^{as seems} observed,

This value of U, ^of order t, corresponds ^{to a weak}

correlation regime. Even if the basic assumptions ^of

this estimate are accepted, uncertainties in the values of the parameters ^{make at} best the numerical estimate of U only valid within a factor 2. Values of U m 2 eV are often used in the study ^of correlation effects in the d band of transitional metals [42]. They

lead to correlation peaks ⁱⁿ photoemission spectra [43] ^shifted by energies comparable ^{to the 8 eV}

energy shift observed in La2Cu04 ^base compounds [44]. ^{U ~ 1.8 eV} is also the shift in X-ray absorption

spectra of Cu in these compounds, ^{which can} possibly be attributed to Cu+ + + ions [45].

Two final remarks can be made.

The phase diagram suggested ⁱⁿ figure ^{6 for a} possible ^{SDW assumes} that its critical temperature follows the two dimensional mean field one in its variation with doping [18]. ^{This is} reasonable if lattice anisotropy ^{leads to an} Ising model, ^{with a} strong anisotropy perpendicular ^{to the} planes, ^as might ^{be the case} [46]. ^{If the} anisotropy ^{were weak}

or planar, ^{one could} expect that the maximum

stability of the SDW phase ^{occurs at finite} doping (or ^finite concentration _{of oxygen} vacancies), ^be-

cause the magnetic coupling ^between C02 planes ^is

frustrated. Coupling could then be due to local

imperfections ^{such as 0} vacancies, ^{or to values of} Q slightly ^{different from} z: : z:; ^[47]. More exper-

a b

imental work, ^{with a better} characterization of O vacancies is needed here.

A shift of Fermi level due to doping ^{can indeed}

lead to nesting for Q ¥- ^± IT-7T Nesting ^{is how-}

a b

ever less perfect ^{and the} stability ^{of the SDW}

reduced. A secondary maximum of stability ^{of the}

SDW is expected when the Fermi level reaches a van Hove anomaly El, i.e. when x - 2 y ^{~ 0.1} (Fig.’6). ^{It can however be} easily shown that the

formula giving the critical temperature Tc ^then

involves In Tc ^{and not} (In Tc )2 , ^{as for} undoped

material [48]. ^{Thus SDW} should be less stable than

superconductivity in this range where superconduc- tivity ^is actually observed. But the two instabilities could mix if superconductivity ^{is due to electron-}

electron repulsion [12].

References

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